# Properties

 Label 252.4.a.e Level $252$ Weight $4$ Character orbit 252.a Self dual yes Analytic conductor $14.868$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8684813214$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} -7 q^{7} +O(q^{10})$$ $$q + \beta q^{5} -7 q^{7} -13 \beta q^{11} -30 q^{13} + 23 \beta q^{17} -100 q^{19} + 17 \beta q^{23} -97 q^{25} -44 \beta q^{29} -180 q^{31} -7 \beta q^{35} -118 q^{37} -3 \beta q^{41} -412 q^{43} + 54 \beta q^{47} + 49 q^{49} + 54 \beta q^{53} -364 q^{55} -158 \beta q^{59} -378 q^{61} -30 \beta q^{65} + 244 q^{67} + 83 \beta q^{71} + 670 q^{73} + 91 \beta q^{77} + 216 q^{79} + 152 \beta q^{83} + 644 q^{85} -183 \beta q^{89} + 210 q^{91} -100 \beta q^{95} + 574 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 14q^{7} + O(q^{10})$$ $$2q - 14q^{7} - 60q^{13} - 200q^{19} - 194q^{25} - 360q^{31} - 236q^{37} - 824q^{43} + 98q^{49} - 728q^{55} - 756q^{61} + 488q^{67} + 1340q^{73} + 432q^{79} + 1288q^{85} + 420q^{91} + 1148q^{97} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
0 0 0 −5.29150 0 −7.00000 0 0 0
1.2 0 0 0 5.29150 0 −7.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.a.e 2
3.b odd 2 1 inner 252.4.a.e 2
4.b odd 2 1 1008.4.a.bd 2
7.b odd 2 1 1764.4.a.t 2
7.c even 3 2 1764.4.k.t 4
7.d odd 6 2 1764.4.k.w 4
12.b even 2 1 1008.4.a.bd 2
21.c even 2 1 1764.4.a.t 2
21.g even 6 2 1764.4.k.w 4
21.h odd 6 2 1764.4.k.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.a.e 2 1.a even 1 1 trivial
252.4.a.e 2 3.b odd 2 1 inner
1008.4.a.bd 2 4.b odd 2 1
1008.4.a.bd 2 12.b even 2 1
1764.4.a.t 2 7.b odd 2 1
1764.4.a.t 2 21.c even 2 1
1764.4.k.t 4 7.c even 3 2
1764.4.k.t 4 21.h odd 6 2
1764.4.k.w 4 7.d odd 6 2
1764.4.k.w 4 21.g even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(252))$$:

 $$T_{5}^{2} - 28$$ $$T_{11}^{2} - 4732$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-28 + T^{2}$$
$7$ $$( 7 + T )^{2}$$
$11$ $$-4732 + T^{2}$$
$13$ $$( 30 + T )^{2}$$
$17$ $$-14812 + T^{2}$$
$19$ $$( 100 + T )^{2}$$
$23$ $$-8092 + T^{2}$$
$29$ $$-54208 + T^{2}$$
$31$ $$( 180 + T )^{2}$$
$37$ $$( 118 + T )^{2}$$
$41$ $$-252 + T^{2}$$
$43$ $$( 412 + T )^{2}$$
$47$ $$-81648 + T^{2}$$
$53$ $$-81648 + T^{2}$$
$59$ $$-698992 + T^{2}$$
$61$ $$( 378 + T )^{2}$$
$67$ $$( -244 + T )^{2}$$
$71$ $$-192892 + T^{2}$$
$73$ $$( -670 + T )^{2}$$
$79$ $$( -216 + T )^{2}$$
$83$ $$-646912 + T^{2}$$
$89$ $$-937692 + T^{2}$$
$97$ $$( -574 + T )^{2}$$